Abstract

We consider time series data modelling using Hidden Markov Models having an a priori unknown number of hidden states. We show that the Infinite Hidden Markov Model of Beal, Ghahramani & Rasmussen (2002) can be recast in the framework of Hierarchical Dirichlet Processes (HDPs). The HDP framework (Teh, Jordan, Beal & Blei, 2004) considers problems involving related groups of data: each (fixed) group of data is modelled by a DP mixture model, with the common base measure of the DPs being itself distributed according to a global DP. The base measure being discrete w.p.1 ensures that the group DPs share atoms (despite being countably infinite). Teh et al. (2004) present two sampling schemes for posterior inference in the HDP : the Chinese Restaurant Franchise and an auxiliary variable scheme.

We cast sequential data in the grouped data framework by assigning observations to groups, where the groups are indexed by the value of the previous state variable in the sequence; then the current state and its emission distributions define a group-specific mixture model. Thus the hidden state sequence implicitly defines a partition into groups, and induces constraints in the posterior that make the CRF sampling methods proposed quite difficult. We construct an auxiliary variable sampling scheme for the iHMM, present results on some small data sets and consider an interesting extension for language modelling.

Joint work with Yee Whye Teh, Michael I. Jordan and David Blei